 # 1.01 Rational and irrational numbers

Lesson

## What is an irrational number?

A rational number is a number which can be written as a fraction where both the numerator and denominator are integers. An irrational number is a number which cannot be written as a fraction of two integers.

We can write any terminating or repeating decimal as a fraction, therefore these are rational numbers. However, decimals which are neither terminating nor recurring are irrational numbers.

You might be familiar with one irrational number already: $\pi$π. Like all other irrational numbers, $\pi$π really does go on forever without repeating itself. We say therefore that it doesn't terminate, or repeat.

Another number that is famously irrational is $\sqrt{2}$2. In fact, the square root of most numbers are irrational. If a root is irrational it is called a radical. The square roots of perfect squares are rational, $\sqrt{1},\sqrt{4},\sqrt{9},\dots$1,4,9,.

#### Practice question

##### Question 1

Is $\sqrt{35}$35 rational or irrational?

1. Rational

A

Irrational

B

##### QUESTION 2

Is $\sqrt{23}$323 rational or irrational?

1. Rational

A

Irrational

B

But how can we write irrational numbers as decimals?

### Exact forms and approximations

Consider one third. This can be written as a fraction $\frac{1}{3}$13, and as a decimal we know it repeats forever as $0.333333\dots$0.333333. So if we want to do an exact calculation that includes $\frac{1}{3}$13, we should keep it as a fraction throughout the calculation.

If we type $1\div3$1÷​3 into a calculator, and it would show us around $8$8 or $9$9 digits on the screen. This is now an approximation. $0.3333333333$0.3333333333 is a good approximation of $\frac{1}{3}$13, but even this has been rounded to fit on your calculator screen, so it is no longer the exact value.

### Converting from fractions to decimals

To convert from a fraction to a decimal, we can rewrite the fraction as division expression. For example, $\frac{3}{7}$37 is $3$3 divided by $7$7. Then we can use short division, adding extra zeros as required to the numerator.

#### Practice question

##### Question 3

Write the fraction $\frac{5}{9}$59 as a repeating decimal.

Rational and irrational numbers

rational number is a number which can be written as a fraction where both the numerator and denominator are integers.

An irrational number is a number which cannot be written as a fraction of two integers.

A radical is a root which is irrational.

A number has an exact value. In the case of fractions and roots, the exact value must be a fraction or root.

Numbers also have approximations. These are numbers which are close but not equal to the exact value. We usually find approximations by rounding the exact value.

For example, if $\frac{2}{3}$23 is the exact value, then $0.667$0.667 is an approximation.

### Estimating square roots

Previously we have looked at evaluating square roots of perfect squares such as $\sqrt{25}$25 or $\sqrt{361}$361. Most of the time when we take square roots it will not be of a perfect square. We can use calculators or estimate in this case.

This applet can help you to learn the first $20$20 perfect squares from $1$1 to $400$400.

 Created with Geogebra

There are also some ways to approximate square roots without using a calculator directly. One way is to consider the nearest integer value as a way to estimate or check our work.

Estimating square roots

### Outcomes

#### 8.B1.2

Describe, compare, and order numbers in the real number system (rational and irrational numbers), separately and in combination, in various contexts.

#### 8.B1.3

Estimate and calculate square roots, in various contexts.

#### 8.B2.2

Understand and recall commonly used square numbers and their square roots.